2577 ÷ 7 = 368 R1
Alright, let's work through the division of 2577 by 7 step by step.

Step 1:
7 goes into 2 0 times with a remainder of 2.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 2 to get 2.
Bring next digit (5) of the dividend behind the 2 and repeat the process: 25 / 7

Step 2:
7 goes into 25 3 times with a remainder of 4.
Write down 3 as next digit of of the result. 
Result so far: 3
Subtract 21 from 25 to get 4.
Bring next digit (7) of the dividend behind the 4 and repeat the process: 47 / 7

Step 3:
7 goes into 47 6 times with a remainder of 5.
Write down 6 as next digit of of the result. 
Result so far: 36
Subtract 42 from 47 to get 5.
Bring next digit (7) of the dividend behind the 5 and repeat the process: 57 / 7

Step 4:
7 goes into 57 8 times with a remainder of 1.
Write down 8 as next digit of of the result. 
Result so far: 368
Subtract 56 from 57 to get 1.

The final result is 368 with a remainder of 1.